An introduction to topological hyperrings

In this paper, we define topological hyperrings and study their basic concepts which supported by illustrative examples. We show some differences between topological rings and topological hyperrings. Also, by the fundamental relation Γ^∗ , we indicate the role of complete parts (saturated subsets) and complete hyperrings in topological hyperrings and specially we show that if every (closed) open subset is a complete part in a topological complete hyperring then its fundamental ring is a topological ring. Finally, we study the quotient topology induced by Γ^∗ -relation on an associated Krasner hyperring obtained by a ring and show that it is isomorphic to a quotient space of the ring by its ideals.